2. Electronic and ionic conductivity

Learning subject

1. Classes of conductors

2. Mobility and transport number

3. Conductivity

Learning objective

1. To identify electronic and ionic conductors

2. Understanding concepts of mobility and transport number

3. Understanding the concept of conductivity

Heith B. Oldham, Jan C. Myland, Allan M. Bond, Electrochemical Science & Technology:Fundamentals & Applications, Wiley, 2013. (e-book, SNU Library)

Electrochemical Energy Engineering, 2019

1. Classes of conductorsMaterials 1.Conductors Electronic conductors

Ionic conductors

2. Insulators

Conductors: metals

Insulators: plastics, ceramics, gases

No clear cut distinction between conductor and insulator

Typical value of electrical conductivity

S/m x10-2 for S/cm

Material /Sm-1

Ionic conductors

Electronic conductors

Ionic crystals

Solid electrolytes

Strong(liquid) electrolytes

Metals

Semiconductors

Insulators

10-16 – 10-2

10-1 – 103

10-1 – 103

103 – 107

10-3 – 104

<10-10

Material /Sm-1 Charge carriers

Electron pairs

Electrons

Electrons

Electrons

Pi electrons

Pi electrons

K+ and Cl-

H+ and HSO4-

Cations & anions

Electrons and holes

K+ and Cl-

H+ and OH-

Univalent cations

?

Electrical conductivity of various materials (most at 298 K)

Superconductors (low temp)

Ag

Cu

Hg

C (graphite)

Doped polypyrrole

Molten KCl (at 1043 K)

5.2 M H2SO4 (battery acid)

Seawater

Ge

0.1 M KCl

H2O

Typical glass

Teflon, (CF2)n

Vacuum & most gases

6.3 x 107

6.0 x 107

1.0 x 106

4 x 104

6 x 103

217

82

5.2

2.2

1.3

5.7 x 10-6

3 x 10-10

10-15

0

Electronically conductive polymers

Measurement of electrical conductivity

1. Four terminal method: calculation from measured I, , A and x

2. a.c. impedance method

The nature of the charge carriers

1) Electronic conductors: mobile electrons; metals, some inorganic oxides and

sulfides (e.g., PbO2 and Ag2S which are slightly non-stoichiometric),

semiconductors (n-type: electrons, p-type: holes, intrinsic: both), conducting

polymer (pi-electrons), graphite(pi-electrons), organic metals (organic salts, e.g.,

TTF-TCNQ(tetrathiafulvalene tetracyanoquinodimethane, pi-electrons)

• Metals: shared valence electrons with all atoms in solid (delocalized electrons)

high electric and thermal conductivity

cf: insulator vs. conductor: valence band completely filled vs. partially filled

e.g., Diamond (insulator); sp3 orbital (completely filled valence band), Eg: 5.6 eV

Na (alkali metal); 11 electrons (10 filled 1s & 2p, 1 valence electron 3s (half

filled electric conduction using unfilled part of VB)

Alkaline earth metal (divalent, 12 e’s) good conductors because their

valence band overlaps another band

Conductivity of metal increases as temperature lowered or impurities reduced

since low resistance

• Semiconductors: Eg is smaller than insulator (1 ~ 2 eV; relatively small

excitation energy, cf) 1eV = 12000 K = 1240 nm (1.2 m (IR)))

Conductivity of semiconductors increases as temperature & impurity

concentration increased.

• Semimetals; between metals & semiconductors, e.g., graphite planar sheet of

hexagons with weak van der Waals forces (2-dimensional molecule), Eg = 0 (top

energy level of pi()-bonding orbitals (the valence band) is at the same level of

that of the anti-bonding orbital

• Conducting polymer: -electrons

2) Ionic conductors: motion of anions and/or cations; solutions of electrolytes

(salts, acids and bases) in water and other liquids, molten salts, solid ionic

conductors (solid electrolyte)( O2- in ZrO2 at high temperature, Ag+ in RbAg4I5 at

room temperature, fluoride ion holes in EuF2 doped LaF3)

3) Electronic & ionic conductors; plasmas (hot gases, positive ions and free

electrons), sodium metal in liquid ammonia(Na+ cation and solvated electrons),

hydrogen dissolved in Pd metal(hydrogen ions(protons) and electrons)

conductors electronic metals

some inorganic oxides & sulfides

semiconductors n-type

intrinsic

p-type

organic metals

conducting polymers

mixed plasmas

some solids & solutions

ionic solutions of electrolytes

molten salts

solid ionic conductors

doped crystals

Mobilities: conduction from the standpoint of the charge carriers

Electric current = rate at which charge crosses any plane = [number of carriers

per unit volume][cross sectional area][charge on each carrier][average carrier

speed]

I = dq/dt = (NAci)(A)(qi)(i)

i: particular charge carrier, ci; concentration, qi; charge, i; average velocity,

NA; Avogadro’s constant (6.0220 x 1023 mol-1), A; area

zi; charge number = qi/qe where qe (1.6022 x 10-19 C),

e.g., electrons:-1, Mg2+; +2

i fi X d/dx

fi; force exerted on the charge carrier, X; electric field strength

2. Mobility and transport number

mobility of the carrier, ui (m2s-1V-1 unit) = velocity to field ratio (i / X)

i = uiX = - (zi/zi)uid/dx

zi: absolute value of the charge number

ue- of electrons: 6.7 x 10-3 m2s-1V-1 for Ag, less mobile in other metals

mobility of ions in aqueous solution: smaller than the factor of 105 (factor 105

slower); ucu2+o = 5.9 x 10-8 m2s-1V-1 in extremely diluted solution

Current I,

I = -A NAqeziuicid/dx

Faraday constant

F = NAqe = (6.02 x 1023 mol-1)(1.6022 x 10-19 C) = 96485 Cmol-1

is numerically equal to the charge carried by one mole of univalent cations.

(F is large. Small amount of chemicals higher electricity)

If there are several kind of charge carriers,

I = -AFd/dxziuici

i = -Fd/dxziuici

Transport number ti; the fraction of the total current carried by one particular

charge carrier

ti = (ziuici )/(ziuici)

From i = X = -d/dx,

conductivity

= Fziuici

molar ionic conductivity (i); Fui

Ion mobilities at extreme dilution in aqueous solution at 298 K

Ion uo/m2s-1V-1

H+

K+

Ag+

Cu2+

Na+

Li+

OH-

SO42-

Cl-

ClO4-

C6H5COO-

362.5 x 10-9

76.2 x 10-9

64.2 x 10-9

58.6 x 10-9

51.9 x 10-9

40.1 x 10-9

204.8 x 10-9

82.7 x 10-9

79.1 x 10-9

69.8 x 10-9

33.5 x 10-9

cf. ue- of electrons: 6.7 x 10-3 m2s-1V-1 for Ag

Capacitance

parallel conducting plate separated by a narrow gap containing air or insulator

Idt = Q E

Q = -CE

C; capacitance (unit; farads (F) = C/V)

C = -Q/E = A/L

A;cross-section area of the gap, L; width, ; permittivity of the insulator

• Relative permittivity (r) or dielectric constant

air: ~ 1

water: 78 Coulomb interaction energy is reduced by two orders of magnitudes

from its vacuum value

polar molecules: r

refractive index: nr = r1/2 at the frequency

Capacitor; ; current integrator

/0; relative permittivity or dielectric constant

mylar; poly(ethylene glycol terephthalate), (CH2OOCC6H4COOCH2)n

Liquid > solid: large capacitance in electrochemical capacitor (supercapacitor)

Material 1012 /Fm-1 Material 1012 /Fm-1

vacuum (0)

N2(g)

Teflon(s), (CF2)n

CCl4(l)

Polyethene (s)

Mylar (s)

SiO2(s)

Typical glass (s)

C6H5Cl(l)

8.85419

8.85905

18

19.7

20

28

38.1

44

49.8

Neoprene

ClC2H4Cl(l)

CH3OH(l)

C6H5NO2(l)

CH3CN(l)

H2O(l)

HCONH2(l)

TiO2(s)

BaTiO3(s)

58

91.7

288.9

308.3

332

695.4

933

1500

110000

Permittivity of various materials

Electricity flows either by electron motion or ion motion

In both cases,

the intensity of the flow (= current density) electric field strength

i = X = -d/dx

conductivity

= Fziuici

determined by the concentration of charge carriers and their mobilities

one form of Ohm’s law

E = -RI

potential difference across resistor to the current flowing through it

Resistor: dissipate energy

Capacitor: store energy

3. Conductivity

Transference number (or transport number)

The fraction of the current carried by H+ and Cl-: t+ and t-

t+ + t- = 1

∑ti = 1

e.g., Figure above: t+ = 0.8, t- = 0.2

Conductance (S = Ω-1), L = κA/l

conductivity (κ, Scm-1): contribution from all ionic species

ion conc, charge magnitude (|zi|), index of migration velocity (ui)

Mobility (ui): limiting velocity of the ion in an electric field of unit strength

unit: cm2V-1s-1 (cm/s per V/cm)

electric field, E → electric force → counterbalance with frictional drag →

terminal velocity

Electric force = |zi|eE e: electronic charge

Frictional drag (Stokes law) = 6πηrv

η :viscosity of medium, r: ion radius, v: velocity

When the terminal velocity is reached:

ui = v/E = |zi|e/ 6πηr

Conductivity

κ = F∑|zi|uiCi

Transference number for species i = conductivity by i /total conductivity

ti = |zi|uiCi/∑|zj|ujCj

For pure electrolytes (e.g., KCl, CaCl2, HNO3) → equivalent conductivity (Λ)

Λ = κ/Ceq (conductivity per unit concentration of charge)

Ceq: concentration of + (or -) charges = C|z|

Λ = F(u+ + u-) = λ+ + λ-

equivalent ion conductivity, λi = Fui

ti = λi/Λ = ui/(u+ + u-)

- Table: t+ → individual ionic conductivities, λi

- λi, ti depend on concentration of pure electrolyte because interactions between

ions tend to alter mobilities

→ Table : λ0i (extrapolated to infinite dilution) → calculate ti

For pure electrolyte: ti = λi/Λ

For mixed electrolytes: ti = |zi|Ciλi/∑|zj|Cjλj

Solid electrolyte: ions move under electric field without solvent → conductivity

→ batteries, fuel cells, and electrochemical devices